Theorem eqvrelsymb 37989

Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised and distinct variable conditions removed by Peter Mazsa, 2-Jun-2019.)

Hypothesis

Ref	Expression
eqvrelsymb.1	⊢ (𝜑 → EqvRel 𝑅)

Assertion

Ref	Expression
eqvrelsymb	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))

Proof of Theorem eqvrelsymb

Step	Hyp	Ref	Expression
1		eqvrelsymb.1	. . . 4 ⊢ (𝜑 → EqvRel 𝑅)
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → EqvRel 𝑅)
3		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4	2, 3	eqvrelsym 37988	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵𝑅𝐴)
5	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → EqvRel 𝑅)
6		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐵𝑅𝐴)
7	5, 6	eqvrelsym 37988	. 2 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐵)
8	4, 7	impbida 798	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   class class class wbr 5141   EqvRel weqvrel 37573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-refrel 37895  df-symrel 37927  df-trrel 37957  df-eqvrel 37968

This theorem is referenced by:  eqvrelth  37994